# Determinant with Row Multiplied by Constant/Proof 1

## Theorem

Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$.

Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$.

Let $\mathbf B$ be the matrix resulting from one row of $\mathbf A$ having been multiplied by a constant $c$.

Then:

$\map \det {\mathbf B} = c \map \det {\mathbf A}$

That is, multiplying one row of a square matrix by a constant multiplies its determinant by that constant.

## Proof

Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$.

Let $e$ be the elementary row operation that multiplies rows $i$ by the scalar$c$.

Let $\mathbf B = \map e {\mathbf A}$.

Let $\mathbf E$ be the elementary row matrix corresponding to $e$.

$\mathbf B = \mathbf E \mathbf A$
$\map \det {\mathbf E} = c$

Then:

 $\ds \map \det {\mathbf B}$ $=$ $\ds \map \det {\mathbf E \mathbf A}$ Determinant of Matrix Product $\ds$ $=$ $\ds c \map \det {\mathbf A}$ as $\map \det {\mathbf E} = c$

Hence the result.

$\blacksquare$